This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Part 2: Solve the right triangle using trigonometry (adjacent side = 5 cm, = 20^, opposite = x cm, hypotenuse = y cm) Step 1: Use tangent for opposite side. = (x)/(5) 20^ = (x)/(5) x = 5 20^ ≈ 5 × 0.3640 = 1.82\ cm Step 2: Use cosine for hypotenuse. = (5)/(y) y = (5)/( 20^) y ≈ (5)/(0.9397) ≈ 5.32\ cm Step 3: Verify with sine. 20^ = (x)/(y) ≈ (1.82)/(5.32) ≈ 0.3420 Matches 20^. Part 3: Solve the triangle using law of cosines (sides a=6 cm, b=10 cm, C=10^, find side c, then angles) Step 1: Law of cosines for side c. c^2 = a^2 + b^2 - 2ab C c^2 = 6^2 + 10^2 - 2 · 6 · 10 · 10^ c^2 = 36 + 100 - 120 · 0.9848 ≈ 136 - 118.18 = 17.82 c ≈ sqrt(17.82) ≈ 4.22\ cm Step 2: Law of cosines for A. A = (b^2 + c^2 - a^2)/(2bc) A = (10^2 + 4.22^2 - 6^2)/(2 · 10 · 4.22) ≈ (100 + 17.81 - 36)/(84.4) ≈ (81.81)/(84.4) ≈ 0.9695 A ≈ ^-1(0.9695) ≈ 14.1^ Step 3: Find B = 180^ - 10^ - 14.1^ = 155.9^. Part 4: Solve x = (y + 5)/(s) for s in terms of x and y Step 1: Multiply both sides by s. x s = y + 5 Step 2: Isolate s. s = (y + 5)/(x) Part 5: Solve (dx)/(dt) = (dy)/(dt) + 10 for y' = (dy)/(dt) (speeds in m/s) Step 1: Subtract (dy)/(dt) from both sides. (dx)/(dt) - (dy)/(dt) = 10 Step 2: Isolate y'. y' = (dx)/(dt) - 10 Part 6: Measures of central tendency for data (value, frequency): (2, 6), (3, 10) Step 1: Mode is the value with highest frequency. Mode = 3 (frequency 10 > 6) Step 2: Mean x = ( (x · f))/( f). (x · f) = 2 · 6 + 3 · 10 = 12 + 30 = 42 f = 6 + 10 = 16 x = (42)/(16) = (21)/(8) = 2.625 Mean = 2.625 x ≈ 1.82 cm y ≈ 5.32 cm c ≈ 4.22 cm A ≈ 14.1^ B ≈ 155.9^ s = (y+5)/(x) y' = (dx)/(dt) - 10 mode = 3 mean = 2.625